I have the same problem that my touch Id is not working anymore after updating to IOS10, before it worked fine!
Also I get directly the failed screen when I try to register a finger and I get a message after rebooting that it is not possible to configure the touch ID on this iPhone.
I think that the problem is coming when you had a screen replacement before that was not done by an official apple store. I think also that it is a software thing because the error pops up when you reboot. So IOS check on something if the hardware is original/valid and otherwise he give this messages and you are not able to use the touch id anymore.
I think it is a simular check than a not compatible charger that is not working.
So I hope apple wil fix this in a next update from IOS 10.